I've changed my methods since I first made this post. I still recommend these sites as being very thorough, but the single most accurate record of Billboard charts that I've found is
http://song-database.com. It requires a subscription to use more than a tiny bit, but I decided it was worth it ($30 for lifetime - I donated more than that to MB). It's much more accurate (in terms of the historical record) than Billboard.com or even bullfrogspond.com (because bullfrog gets his data, especially the older stuff, from Joel Whitburn, who sometimes combined multiple charts in his accounting). If there's a discrepancy, I usually go with what song-database says. The only exception is for some singles where both sides charted, and Billboard didn't distinguish, but song-database does.
However, bullfrogspond.com has a downloadable spreadsheet, which was recently updated to include stats through Oct 25, 2014. I've taken that and customized it to my purposes, and I'm using a combination of billboard.com and song-database.com (for everything after the Top 100/Hot 100 was begun), and hitsofalldecades.com (for 1940-1955) to corroborate (and sometimes add) the data. The latter site has a few issues that crop up, too, (notably that he cuts many early charts off at 20 even if they were longer) but it's reasonably consistent and accurate in its reporting.
Also, if the data elsewhere seems really shaky, I can check the source! Google Books has almost all of Billboard Magazine since 1942 available. I've caught a few of Joel Whitburn's inaccuracies that had been propagated even as far as Wikipedia by going to the original source. You can find it here:
https://books.google.com/books?id=N_YDAAAAMBAJAm I going way overboard with the level of detail and accuracy in my library? Probably. But I'm having fun!